Optimal. Leaf size=38 \[ -\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {3488} \[ -\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 1.00 \[ -\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 d (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 76, normalized size = 2.00 \[ \frac {2 \, {\left (-i \, a e^{\left (3 i \, d x + 3 i \, c\right )} - i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.20, size = 76, normalized size = 2.00 \[ -\frac {2 \left (i \cos \left (d x +c \right )-\sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (d x +c \right )\right ) a}{3 d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 76, normalized size = 2.00 \[ -\frac {2 i \, a^{\frac {3}{2}} {\left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}}{3 \, d e^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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